Everything I know about Exponents

kaitlyns2016 on

1. Represent repeated multiplication with exponents

4^5 = 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 An exponent is just the amount of times you would multiply the base, and if you wrote it out, you would get a repeated multiplication equation.

2. Describe how powers represent repeated multiplication

An exponent ( the small number ) represents the number of times you multiply the base ( the big number ), for example 3 to the 4 would equal 3•3•3•3, not 3•4 3^4 = 3 \cdot 3 \cdot 3 \cdot 3

3. Demonstrate the difference between the exponent and the base by building models of a given power

The base is equal to the amount of side length on the model and the exponent is equal to the amount of sidfullsizerender-3fullsizerender-4es the model has. For example 2 to the 3rd would be represented as a cube of the side length of 2, and 5 squared would be represented as a 5•5 square model. 2^3 = 2 \cdot 2 \cdot 2 5^2 = 5 \cdot 5

4. Demonstrate the difference between 2 given powers in which the exponent and the base are interchanged by using repeated multiplication 

if you had 2^3 and 3^2 , even though they are the same numbers, its not the same rule applied to multiplication in which the answer will be the same no matter what order the numbers come in. They are 2 separate equations. 2^3 is 2•2•2 which equals 8, and 3^2 is 3•3 which equals 9.

5.  Evaluate powers with integral bases (excluding base 0) and whole number exponents

evaluating powers with integral bases and whole number exponents can be expressed as repeated multiplication, in which you multiply the base the amount of times the exponent declares. 5^3 = 5 \cdot 5 \cdot 5 , 4^0 = 1 , 354^2 = 354 \cdot 354

6. Explain the role of parentheses in powers by evaluating a given set of powers

\langle -2 \rangle ^3 is different then \langle -2^3 \rangle and -2^3 , which are the same. If the negative is inside the brackets but the exponent is outside, it basically means the exponent is repeated multiplication of whatever is inside the brackets. That would be represented as -2•-2•-2. If the negative sign and the exponent were inside the brackets, it would mean the negative sign would become the coefficient -1 and then you would only put the base to the exponent. It would then be represented as -1•2•2•2. That is also the same as no brackets at all. If the exponent is outside of the brackets and the negative sign is outside of the brackets, that’s also represented as -1•2•2•2.

 7. Explain the exponent laws for multiplying and dividing powers with the same base

Multiplying powers with the same base means you would just add the exponents of the numbers. Say you have the equation 5^3 \cdot 5^2 . If you were to write that out as repeated multiplication you would get (5•5•5)(5•5), which is the same as 5•5•5•5•5, which is the same as 5^5 . So basically if you are multiplying 2 powers with the same base you just add the exponents. 5^{3+2}

And that’s the same with division. If you are dividing 2 powers with the same base, you would subtract the exponents, because division is repeated subtraction. 4^7 \div 4^3 is the same as \frac{4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4}{4 \cdot 4 \cdot 4} and then you can cross out the 4’s that cancel each other out, so you would be left with 4 \cdot 4 \cdot 4 \cdot 4 . That’s the same as just doing 4^{7-3} .

8. Explain the exponent laws for raising a product and quotient to an exponent

when you raise a power to an exponent, you just multiply the exponents. If you had \langle 3^3 \rangle ^2 it’s the same as (3•3•3)(3•3•3), which is the same as 3^{3 \cdot 2} , and 3^6

9. Explain the law for powers with an exponent of zero

Anything to the power of zero equals 1. This is because If you put it like 4^3 \div 4^3 would equal \frac{4^3}{4^3} and if you use BEDMAS you would get \frac{64}{64} which equals 1.

10. Use patterns to show that a power with an exponent of zero is equal to one

if you look at a pattern of powers such as 2^4 = 16, 2^3 = 8, 2^2 = 4, 2^1 = 2, 2^0 = 1, you can see that each time the exponent decreases by 1, the answer divides by the base, so you can see that 2^0 would have to equal 1.

11. Explain the law for powers with negative exponents

if you have a power with a negative exponent, it’s the same as having the power with the exponent positive, in a fraction under 1. If you have 2^{-5} , it can be displayed as \frac{2^{-5}}{1} , and to make it a positive exponent, you just switch the numerator and the denominator which would be \frac{1}{2^5}

12. Use patterns to explain the negative exponent law

if you have the examples 2^2 = 4 , 2^1 = 2 , 2^0 = 1 , you can see that if you continue the pattern of when you decrease the exponent by 1 then you divide the answer by the base, you would get 2^{-1} = \frac{1}{2} , then 2^{-2} = \frac{1}{4} , and to validate the rule you can see that you get the same answers using the negative exponents rule. 2^{-2} = \frac{2^{-2}}{1} = \frac{1}{2^2} = \frac{1}{4}

13. I can apply the exponent laws to powers with both integral and variable bases

you can apply all the exponent laws to both integral ad variable bases, because no matter what the base is, the exponent laws only concern the exponent. b^3 \cdot b^4 = b^7 c^6 \div c^2 = c^4 \langle c^2 \rangle ^3 = c^6 c^{-4} = \frac{1}{c^4} a^0 =1

14. I can identify the error in a simplification of an expression involving powers.

Many people may make the mistake of using the exponent laws on equations that do not apply to them, Such as some people may use the product and quotient law for addition and subtraction questions with the same base, but those rules only apply to multiplication and division questions with the same base. Also for the power to the power law, you have to make sure to multiply the exponents, but don’t multiply the base. For the base you would just solve it to that power.\langle 5^2 \rangle ^3 = 5^5 is wrong. \langle 5^2 \rangle ^3 you would multiply the 2 and the 3, and then solve the base to the product of the exponents 5^6 = 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 = 15625

15. Use order of operations on expressions with powers

If you are solving expressions with powers and you can apply the exponent laws, you would do those, but if you cannot use the exponent laws then you would solve using BEDMAS.

16. Determine the sum and difference of two powers

to solve an addition or subtraction equation with exponents, there are no exponent laws so you would just use BEDMAS to solve the powers. 3^3 + 2^4 = 3 \cdot 3 \cdot 3 + 2 \cdot 2 \cdot 2 \cdot 2 = 27 + 16 = 43

17. Identify the error in applying the order of operations in an incorrect solution

Some people may make the mistake of using BEDMAS when you could just use the exponent laws, or vice versa. 5^2 + 5^3 could be commonly mistaken to be solved like the exponent law for multiplying with the same base but it isn’t the same because when you add the 2 powers its not repeated multiplication. Instead you would solve each power separately and then add them.

18. Use powers to solve measurement problems

There are lots of different ways to use powers to solve measurement problems. All of them use exponents to solve measimg_8888urement problems by using models to represent each measurement and exponent. If you had a square in a square, and you wanted to find the remaining shaded area, you would first solve for the squares by squaring the side length, then simply subtract the smaller square from the larger one. You would do this because there are no exponent laws to use for addition and subtraction so you would just use BEDMAS. The same rules apply for cubed objects, just instead of squaring the side length, you would cube it.

If you wanted to img_8887find the volume and/or surface area of a cube you could also use exponents to help solve the problem. Since a cube consists of all equal size, to find the volume, which is length • width • height, you would use the equation 5^3 since all the sides equal the length of 5. To find surface area, you multiply the number of sides of a cube (always 6) by the number of squares on 1 side (theimg_8889 side length squared).

To find the length of a shaded square using Pythagorean Theorem, you would square both of the given side lengths, add them together, then square root the answer to get the length.

19. Use powers to solve growth problems

You can use powers to solve growth problems because most growth problems are about doubling, tripling, quadrupling, etcetera, and since that is a repetition of multiplication, it can be represented as an exponent. Say you have the question “a colony of bacteria triples every hour. You have 10 bacteria. How many will you have after 1 hour? 2?” To solve this question first you would triple the 10 bacteria to get the amount needed for the first hour, which would be 300. Knowing that you would have 300 bacteria after 1 hour, you know you would triple that same amount again to get the amount for the second hour. The formula would look like this. 10•3 (1st hour) •3 (2nd hour)… and so on. An easier way to do this is by using an exponent to represent the amount for each hour. For 2 hours you would have 10 \cdot 3^2 for 3 hours 10 \cdot 3^3 for 24 hours 10 \cdot 3^{24}

20. Any other exponent related questions

when working with powers and exponents, always be sure to know all the laws and tricks. solving exponent equations become a lot simpler when you apply the laws and tricks to them. Some good things to know when working with exponents is that you should always know what order to solve the equation in. Its BEDMAS, but it becomes different when you use the tricks. First you would do the power law, then product or quotient law (in order), then the negative exponent law, and then solve with the remaining information, which means simplifying fractions or solving exponents or whatever gets you to the lowest terms possible.

4 thoughts on “Everything I know about Exponents”

  1. Hi, Kaitlyn. This is Jason commenting on your post.

    All in all, your post looked very good, bit better than mine also. You were very descriptive, made sure you had everything included to ensure the reader would understand your words, used simple vocabulary, and made the equations pop out. Your post as a whole evidently shows me that you know what you’re talking about and obviously would have no problem telling people without any notes, sheets, just with what you know. You had all the info you needed without it being all over the place. Although, on question 13 when you have to write how to use the variable and the integral ones, you didn’t write anything about when the base is an integer. Maybe adding something like {-2}^5 = -32, explaining the trick to doing the negative bases like when the exponent is an odd number, the answer is negative and vice versa
    Great job 🙂
    -Jason S

    Reply

  2. Hi Kaitlyn, this is a comment from Keith Shaw regarding your blog post.

    Overall the post is well written, easy to follow and demonstrates that you have a good understanding of exponents. Your post provided a good refresher to my own knowledge of exponents and even explained a few things that hadn’t occurred to me before such as the proof for why any number raised to the power 0 results in 1.

    In section 9, I would recommend that you add one more step to explain where the equation 4^3 / 4^3 came from as I found myself having to stop and think about this for a minute.

    Great job and I look forward to reading more of your blog posts,
    -Keith

    Reply

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